✦ Exam Vault: Chapter-Wise PYQs (2015-2026) | Chapter 9: Applications Of Trigonometry

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✦ NCF 2023 ALIGNED • CBSE CLASS X ✦

Chapter 9: Some Applications of Trigonometry

CBSE Official PYQs (2015-2026) • Mega Set

✨ Explore Interactive TLMs, Worksheets & Visual Notes at: www.akashmaths.online

📅 CBSE 2026 (30/2/1)⭐ 1 Mark📝 MCQ

Q1. From a point on the ground, which is \(60\text{ m}\) away from the foot of a vertical tower, the angle of elevation of the top of the tower is found to be \(45^\circ\). The height of the tower (in metres) is :

(A) \(30\sqrt{3}\text{ m}\)

(B) \(60\text{ m}\)

(C) \(60\sqrt{3}\text{ m}\)

(D) \(30\text{ m}\)

Correct Answer: (B) \(60\text{ m}\)

📅 CBSE 2025 (30/1/1)⭐ 1 Mark📝 MCQ

Q2. If the height of a vertical pole is equal to the length of its shadow on the ground, then the angle of elevation of the sun is :

(A) \(30^\circ\)

(B) \(45^\circ\)

(C) \(60^\circ\)

(D) \(90^\circ\)

Correct Answer: (B) \(45^\circ\)

📅 CBSE 2024 (30/1/2)⭐ 1 Mark📝 MCQ

Q3. A ladder playing against a vertical wall makes an angle of \(60^\circ\) with the ground. If the foot of the ladder is \(3.5\text{ m}\) away from the wall, then the length of the ladder is :

(A) \(7\text{ m}\)

(B) \(3.5\text{ m}\)

(C) \(7\sqrt{3}\text{ m}\)

(D) \(4\text{ m}\)

Correct Answer: (A) \(7\text{ m}\)

📅 CBSE 2023 (30/3/1)⭐ 1 Mark📝 MCQ

Q4. The angle of depression of a car parked on the road from the top of a \(75\text{ m}\) high tower is \(30^\circ\). The distance of the car from the base of the tower (in metres) is :

(A) \(25\sqrt{3}\)

(B) \(50\sqrt{3}\)

(C) \(75\sqrt{3}\)

(D) \(75\)

Correct Answer: (C) \(75\sqrt{3}\)

📅 CBSE 2020 (30/1/1)⭐ 1 Mark📝 MCQ

Q5. If the angle of elevation of the top of a tower from a point at a distance of \(100\text{ m}\) from its foot is \(60^\circ\), then the height of the tower is :

(A) \(50\sqrt{3}\text{ m}\)

(B) \(\frac{100}{\sqrt{3}}\text{ m}\)

(C) \(100\sqrt{3}\text{ m}\)

(D) \(100\text{ m}\)

Correct Answer: (C) \(100\sqrt{3}\text{ m}\)

📅 CBSE 2017 (30/1)⭐ 1 Mark📝 MCQ

Q6. A pole \(6\text{ m}\) high casts a shadow \(2\sqrt{3}\text{ m}\) long on the ground, then the sun's elevation is :

(A) \(60^\circ\)

(B) \(45^\circ\)

(C) \(30^\circ\)

(D) \(90^\circ\)

Correct Answer: (A) \(60^\circ\)

📅 CBSE 2025 (30/1/2)⭐ 1 Mark📝 MCQ

Q7. If the length of the shadow of a tower is \(\sqrt{3}\) times the height of the tower, then the angle of elevation of the sun is :

(A) \(45^\circ\)

(B) \(30^\circ\)

(C) \(60^\circ\)

(D) \(90^\circ\)

Correct Answer: (B) \(30^\circ\)

📅 CBSE 2024 (30/2/1)⭐ 1 Mark📝 MCQ

Q8. A kite is flying at a height of \(60\text{ m}\) above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is \(60^\circ\). Length of the string (in metres) is :

(A) \(40\sqrt{3}\)

(B) \(30\sqrt{3}\)

(C) \(20\sqrt{3}\)

(D) \(80\)

Correct Answer: (A) \(40\sqrt{3}\)

📅 CBSE 2026 (30/2/2)⭐ 1 Mark📝 MCQ

Q9. The ratio of the length of a vertical rod and its shadow is \(1 : \sqrt{3}\). The angle of elevation of the sun is :

(A) \(30^\circ\)

(B) \(45^\circ\)

(C) \(60^\circ\)

(D) \(90^\circ\)

Correct Answer: (A) \(30^\circ\)

📅 CBSE 2019 (30/1/1)⭐ 1 Mark📝 MCQ

Q10. A tower stands vertically on the ground. From a point on the ground, which is \(15\text{ m}\) away from the foot of the tower, the angle of elevation of the top of the tower is found to be \(60^\circ\). Height of the tower is :

(A) \(15\sqrt{3}\text{ m}\)

(B) \(15\text{ m}\)

(C) \(10\sqrt{3}\text{ m}\)

(D) \(\frac{15}{\sqrt{3}}\text{ m}\)

Correct Answer: (A) \(15\sqrt{3}\text{ m}\)

📅 CBSE 2023 (30/3/2)⭐ 1 Mark📝 MCQ

Q11. If the angle of depression of an object on the ground from the top of a \(100\text{ m}\) high cliff is \(45^\circ\), then the distance of the object from the foot of the cliff is :

(A) \(50\sqrt{2}\text{ m}\)

(B) \(100\text{ m}\)

(C) \(100\sqrt{3}\text{ m}\)

(D) \(50\text{ m}\)

Correct Answer: (B) \(100\text{ m}\)

📅 CBSE 2020 (30/1/3)⭐ 1 Mark📝 MCQ

Q12. The shadow of a \(5\text{ m}\) vertical pole is \(2\text{ m}\) long. At the same time, the shadow of a chimney is \(20\text{ m}\) long. The height of the chimney is :

(A) \(10\text{ m}\)

(B) \(20\text{ m}\)

(C) \(50\text{ m}\)

(D) \(40\text{ m}\)

Correct Answer: (C) \(50\text{ m}\)

📅 CBSE 2025 (30/1/3)⭐ 1 Mark📝 MCQ

Q13. An observer \(1.5\text{ m}\) tall is \(28.5\text{ m}\) away from a chimney. The angle of elevation of the top of the chimney from her eyes is \(45^\circ\). The height of the chimney is :

(A) \(30\text{ m}\)

(B) \(28.5\text{ m}\)

(C) \(1.5\text{ m}\)

(D) \(27\text{ m}\)

Correct Answer: (A) \(30\text{ m}\)

📅 CBSE 2024 (30/2/2)⭐ 1 Mark📝 MCQ

Q14. From a point on the bridge across a river, the angles of depression of the banks on opposite sides of the river are \(30^\circ\) and \(45^\circ\), respectively. If the bridge is at a height of \(3\text{ m}\) from the banks, then the width of the river is :

(A) \(3(\sqrt{3}+1)\text{ m}\)

(B) \(3(\sqrt{3}-1)\text{ m}\)

(C) \(3\sqrt{3}\text{ m}\)

(D) \(6\text{ m}\)

Correct Answer: (A) \(3(\sqrt{3}+1)\text{ m}\)

📅 CBSE 2024 (30/2/3)⭐ 1 Mark📝 MCQ

Q15. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of \(30^\circ\). The angle of depression of the car when it is closer is noted. If the tower is \(50\text{ m}\) high, how far is the car from the tower's base when the angle of depression is \(60^\circ\)?

(A) \(\frac{50}{\sqrt{3}}\text{ m}\)

(B) \(50\sqrt{3}\text{ m}\)

(C) \(25\text{ m}\)

(D) \(100\text{ m}\)

Correct Answer: (A) \(\frac{50}{\sqrt{3}}\text{ m}\)

📅 CBSE 2026 (30/3/1)⭐ 1 Mark📝 MCQ

Q16. The angle of elevation of the top of a building from the foot of the tower is \(30^\circ\) and the angle of elevation of the top of the tower from the foot of the building is \(60^\circ\). If the tower is \(50\text{ m}\) high, then the height of the building is :

(A) \(25\text{ m}\)

(B) \(16\frac{2}{3}\text{ m}\)

(C) \(50\sqrt{3}\text{ m}\)

(D) \(25\sqrt{3}\text{ m}\)

Correct Answer: (B) \(16\frac{2}{3}\text{ m}\)

📅 CBSE 2016 (30/1)⭐ 1 Mark📝 MCQ

Q17. The angle of elevation of the top of a tower from two points at distances of \(a\) and \(b\) (\(a > b\)) from its foot and in the same straight line with it are complementary. The height of the tower is :

(A) \(\sqrt{ab}\)

(B) \(ab\)

(C) \(\sqrt{\frac{a}{b}}\)

(D) \(a^2b^2\)

Correct Answer: (A) \(\sqrt{ab}\)

📅 CBSE 2025 (30/1/1)⭐ 1 Mark📝 MCQ

Q18. From a balloon vertically above a straight road, the angles of depression of two consecutive stones on the same side of the road are found to be \(45^\circ\) and \(60^\circ\). The height of the balloon (in km) is :

(A) \(\frac{\sqrt{3}+1}{2}\)

(B) \(\frac{\sqrt{3}-1}{2}\)

(C) \(\frac{3+\sqrt{3}}{2}\)

(D) \(\frac{3-\sqrt{3}}{2}\)

Correct Answer: (C) \(\frac{3+\sqrt{3}}{2}\)

📅 CBSE 2025 (30/2/2)⭐ 1 Mark📝 MCQ

Q19. A circular park is surrounded by a road of uniform width. An observer at the top of a tower at the center of the park sees a car moving at uniform speed on the road. The angle of depression of the car changes. If the tower is \(h\) metres high and the angles of depression are \(45^\circ\) and \(30^\circ\), then the width of the road is :

(A) \(h(\sqrt{3}-1)\text{ m}\)

(B) \(h(\sqrt{3}+1)\text{ m}\)

(C) \(h\sqrt{3}\text{ m}\)

(D) \(h(\sqrt{2}-1)\text{ m}\)

Correct Answer: (A) \(h(\sqrt{3}-1)\text{ m}\)

📅 CBSE 2026 (30/2/2)⭐ 1 Mark📝 MCQ

Q20. If the angle of elevation of the top of a tower from a point on the ground is \(30^\circ\) and on walking \(20\text{ m}\) towards the tower, the angle of elevation becomes \(60^\circ\), then the height of the tower is :

(A) \(10\text{ m}\)

(B) \(10\sqrt{3}\text{ m}\)

(C) \(20\sqrt{3}\text{ m}\)

(D) \(15\text{ m}\)

Correct Answer: (B) \(10\sqrt{3}\text{ m}\)

📅 CBSE 2026 (30/2/1)⭐ 1 Mark📝 MCQ

Q21. A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height \(h\). At a point on the plane, the angles of elevation of the bottom and top of the flagstaff are \(\alpha\) and \(\beta\) respectively. The height of the tower is :

(A) \(\frac{h \tan \alpha}{\tan \beta - \tan \alpha}\)

(B) \(\frac{h \tan \beta}{\tan \beta - \tan \alpha}\)

(C) \(\frac{h \tan \alpha}{\tan \alpha + \tan \beta}\)

(D) \(\frac{h \tan \beta}{\tan \alpha + \tan \beta}\)

Correct Answer: (A) \(\frac{h \tan \alpha}{\tan \beta - \tan \alpha}\)

📅 CBSE 2024 (30/3/2)⭐ 1 Mark📝 MCQ

Q22. The angle of elevation of the top of a hill from the foot of a tower is \(60^\circ\) and the angle of elevation of the top of the tower from the foot of the hill is \(30^\circ\). If the tower is \(50\text{ m}\) high, then the height of the hill is :

(A) \(150\text{ m}\)

(B) \(100\text{ m}\)

(C) \(50\text{ m}\)

(D) \(200\text{ m}\)

Correct Answer: (A) \(150\text{ m}\)

📅 CBSE 2018 (30/1)⭐ 1 Mark📝 MCQ

Q23. A bridge across a river makes an angle of \(45^\circ\) with the river bank. If the length of the bridge across the river is \(150\text{ m}\), then the width of the river is :

(A) \(75\text{ m}\)

(B) \(75\sqrt{2}\text{ m}\)

(C) \(150\sqrt{2}\text{ m}\)

(D) \(100\text{ m}\)

Correct Answer: (B) \(75\sqrt{2}\text{ m}\)

📅 CBSE 2024 (30/2/3)⭐ 1 Mark📝 MCQ

Q24. If the angle of elevation of a cloud from a point \(h\) metres above a lake is \(\alpha\) and the angle of depression of its reflection in the lake is \(\beta\), then the height of the cloud is represented by :

(A) \(\frac{h(\tan \beta + \tan \alpha)}{\tan \beta - \tan \alpha}\)

(B) \(\frac{h(\tan \beta - \tan \alpha)}{\tan \beta + \tan \alpha}\)

(C) \(\frac{h(\tan \alpha + \tan \beta)}{\tan \alpha - \tan \beta}\)

(D) \(\frac{h(\tan \alpha - \tan \beta)}{\tan \alpha + \tan \beta}\)

Correct Answer: (A) \(\frac{h(\tan \beta + \tan \alpha)}{\tan \beta - \tan \alpha}\)

📅 CBSE 2025 (30/1/3)⭐ 1 Mark📝 MCQ

Q25. A student observes that the angle of elevation of the top of a tower is \(30^\circ\). After walking some distance towards the tower on a horizontal line, the angle of elevation becomes \(45^\circ\). If the tower's height is \(40\text{ m}\), what is the distance walked by the student?

(A) \(40(\sqrt{3}-1)\text{ m}\)

(B) \(40(\sqrt{3}+1)\text{ m}\)

(C) \(40\sqrt{3}\text{ m}\)

(D) \(20\sqrt{3}\text{ m}\)

Correct Answer: (A) \(40(\sqrt{3}-1)\text{ m}\)

📅 CBSE 2026 (30/2/1)⭐ 1 Mark📝 MCQ

Q26. Assertion (A): The height of a building is \(20\text{ m}\). When the sun's altitude is \(60^\circ\), the shadow is shorter than when the sun's altitude is \(30^\circ\).
Reason (R): As the angle of elevation of the sun increases, the length of the shadow of a vertical tower decreases.

(A) Both (A) and (R) are true and (R) is correct explanation.

(B) Both (A) and (R) are true but (R) is not correct explanation.

(C) (A) is true but (R) is false.

(D) (A) is false but (R) is true.

Correct Answer: (A) Both (A) and (R) are true and (R) is the correct explanation of (A).

📅 CBSE 2023 (30/3/1)⭐ 1 Mark📝 MCQ

Q27. From the top of a \(60\text{ m}\) high lighthouse, the angles of depression of two ships are \(30^\circ\) and \(45^\circ\). If one ship is exactly behind the other on the same side of the lighthouse, the distance between the two ships is :

(A) \(60(\sqrt{3}-1)\text{ m}\)

(B) \(60(\sqrt{3}+1)\text{ m}\)

(C) \(60\sqrt{3}\text{ m}\)

(D) \(120\text{ m}\)

Correct Answer: (A) \(60(\sqrt{3}-1)\text{ m}\)

📅 CBSE 2019 (30/1/3)⭐ 1 Mark📝 MCQ

Q28. A vertical pole of length \(7.5\text{ m}\) casts a shadow \(5\text{ m}\) long on the ground and at the same time a tower casts a shadow \(24\text{ m}\) long. The height of the tower is :

(A) \(36\text{ m}\)

(B) \(40\text{ m}\)

(C) \(32\text{ m}\)

(D) \(28\text{ m}\)

Correct Answer: (A) \(36\text{ m}\)

📅 CBSE 2022 (30/2/3)⭐ 1 Mark📝 MCQ

Q29. If the angle of depression of a boat from the top of a bridge of height \(20\text{ m}\) is \(30^\circ\), how far is the boat from the bridge (horizontal distance)?

(A) \(20\text{ m}\)

(B) \(20\sqrt{3}\text{ m}\)

(C) \(\frac{20}{\sqrt{3}}\text{ m}\)

(D) \(40\text{ m}\)

Correct Answer: (B) \(20\sqrt{3}\text{ m}\)

📅 CBSE 2022 (30/3/1)⭐ 1 Mark📝 MCQ

Q30. The angle of elevation of the top of a vertical tower from a point on the ground is \(60^\circ\). From another point \(10\text{ m}\) vertically above the first, the angle of elevation is \(45^\circ\). The height of the tower is :

(A) \(5(\sqrt{3}+3)\text{ m}\)

(B) \(5(3\sqrt{3}+1)\text{ m}\)

(C) \(5(3+\sqrt{3})\text{ m}\)

(D) \(10\sqrt{3}\text{ m}\)

Correct Answer: (A) \(5(\sqrt{3}+3)\text{ m}\)

📅 CBSE 2025 (30/1/1)⭐ 1 Mark📝 MCQ

Q31. If the angle of elevation of a vertical tower from a point on the ground is \(45^\circ\), and the distance of the point from the tower's base is \(25\text{ m}\), then the tower's height is :

(A) \(25\text{ m}\)

(B) \(12.5\text{ m}\)

(C) \(25\sqrt{2}\text{ m}\)

(D) \(50\text{ m}\)

Correct Answer: (A) \(25\text{ m}\)

📅 CBSE 2024 (30/2/2)⭐ 1 Mark📝 MCQ

Q32. From the top of a \(7\text{ m}\) high building, the angle of elevation of the top of a cable tower is \(60^\circ\) and the angle of depression of its foot is \(45^\circ\). The height of the tower is :

(A) \(7(\sqrt{3}+1)\text{ m}\)

(B) \(7(\sqrt{3}-1)\text{ m}\)

(C) \(7\sqrt{3}\text{ m}\)

(D) \(14\text{ m}\)

Correct Answer: (A) \(7(\sqrt{3}+1)\text{ m}\)

📅 CBSE 2024 (30/1/1)⭐ 1 Mark📝 MCQ

Q33. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle \(30^\circ\) with it. The distance between the foot of the tree to the point where the top touches the ground is \(8\text{ m}\). Height of the tree is :

(A) \(8\sqrt{3}\text{ m}\)

(B) \(\frac{8}{\sqrt{3}}\text{ m}\)

(C) \(24\text{ m}\)

(D) \(16\text{ m}\)

Correct Answer: (A) \(8\sqrt{3}\text{ m}\)

📅 CBSE 2023 (30/3/1)⭐ 1 Mark📝 MCQ

Q34. From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a \(20\text{ m}\) high building are \(45^\circ\) and \(60^\circ\) respectively. The height of the tower is :

(A) \(20(\sqrt{3}-1)\text{ m}\)

(B) \(20(\sqrt{3}+1)\text{ m}\)

(C) \(20\sqrt{3}\text{ m}\)

(D) \(40\text{ m}\)

Correct Answer: (A) \(20(\sqrt{3}-1)\text{ m}\)

📅 CBSE 2024 (30/2/3)⭐ 1 Mark📝 MCQ

Q35. A \(1.5\text{ m}\) tall boy is standing at some distance from a \(30\text{ m}\) tall building. The angle of elevation from his eyes to the top of the building increases from \(30^\circ\) to \(60^\circ\) as he walks towards the building. The distance he walked towards the building is :

(A) \(19\sqrt{3}\text{ m}\)

(B) \(28.5\sqrt{3}\text{ m}\)

(C) \(15\sqrt{3}\text{ m}\)

(D) \(19\text{ m}\)

Correct Answer: (A) \(19\sqrt{3}\text{ m}\)

📅 CBSE 2022 (30/3/1)⭐ 2 Marks📝 SA-I

Q36. An observer on a tower sees a car moving at uniform speed towards its base at an angle of depression of \(30^\circ\). After 10 seconds, the angle of depression becomes \(60^\circ\). Find the time taken by the car to reach the foot of the tower.

Solution: \(5\) seconds

📅 CBSE 2025 (30/1/2)⭐ 2 Marks📝 SA-I

Q37. A \(6\text{ m}\) high vertical pole casts a shadow of \(4\text{ m}\) long on the ground, and at the same time a tower casts a shadow of \(28\text{ m}\) long. Find the height of the tower.

Solution: \(42\text{ m}\)

📅 CBSE 2026 (30/2/1)⭐ 2 Marks📝 SA-I

Q38. From a point on the ground, the angle of elevation of the top of a tower is \(30^\circ\). On walking \(40\text{ m}\) towards the tower, the angle of elevation becomes \(60^\circ\). Find the height of the tower.

Solution: \(20\sqrt{3}\text{ m}\)

📅 CBSE 2022 (30/2/3)⭐ 2 Marks📝 SA-I

Q39. A kite is flying at a height of \(75\text{ m}\) from the level ground, attached to a string inclined at \(60^\circ\) to the horizontal. Find the length of the string, assuming there is no slack in it.

Solution: \(50\sqrt{3}\text{ m}\)

📅 CBSE 2023 (30/3/2)⭐ 2 Marks📝 SA-I

Q40. If the shadow of a tower is \(30\text{ m}\) longer when the sun's altitude is \(30^\circ\) than when it is \(60^\circ\), find the height of the tower.

Solution: \(15\sqrt{3}\text{ m}\)

📅 CBSE 2024 (30/2/1)⭐ 2 Marks📝 SA-I

Q41. From the top of a \(50\text{ m}\) high building, the angle of depression of a car on the ground is observed to be \(30^\circ\). Find the distance of the car from the building.

Solution: \(50\sqrt{3}\text{ m}\)

📅 CBSE 2024 (30/1/2)⭐ 2 Marks📝 SA-I

Q42. A vertical flagstaff stands on the top of a building. The angle of elevation of the top of the flagstaff from a point on the ground is \(45^\circ\) and the height of the building is \(10\text{ m}\). If the height of the flagstaff is \(5\text{ m}\), find the angle of elevation of the bottom of the flagstaff.

Solution: \(30^\circ\)

📅 CBSE 2018 (30/1)⭐ 2 Marks📝 SA-I

Q43. The angle of elevation of the top of a tower from a point on the ground \(30\text{ m}\) away from its foot is \(30^\circ\). Find the height of the tower.

Solution: \(10\sqrt{3}\text{ m}\)

📅 CBSE 2025 (30/1/3)⭐ 2 Marks📝 SA-I

Q44. From a point on the ground, the angle of elevation of the top of a chimney is \(60^\circ\). If the chimney is \(150\text{ m}\) high, how far is the observer from its foot?

Solution: \(50\sqrt{3}\text{ m}\)

📅 CBSE 2023 (30/3/2)⭐ 2 Marks📝 SA-I

Q45. A technician is climbing up a \(20\text{ m}\) high pole to fix an electrical fault. If the ladder makes an angle of \(60^\circ\) with the ground, what is the length of the ladder?

Solution: \(\frac{40}{\sqrt{3}}\text{ m}\)

📅 CBSE 2020 (30/1/2)⭐ 2 Marks📝 SA-I

Q46. If a tower of height \(100\text{ m}\) casts a shadow \(100\sqrt{3}\text{ m}\) long, what is the angle of elevation of the sun?

Solution: \(30^\circ\)

📅 CBSE 2024 (30/2/2)⭐ 2 Marks📝 SA-I

Q47. A straight road leads to the foot of a tower. From the top of the tower, a car is spotted at an angle of depression of \(30^\circ\). If the car is \(150\text{ m}\) away from the foot of the tower, find the height of the tower.

Solution: \(50\sqrt{3}\text{ m}\)

📅 CBSE 2023 (30/3/1)⭐ 2 Marks📝 SA-I

Q48. Two poles of equal heights are standing opposite each other on either side of the road, which is \(80\text{ m}\) wide. The angle of elevation of the top of the poles are \(60^\circ\) and \(30^\circ\) respectively. Find the distance of the point on the road from the poles.

Solution: Distance from first pole = \(20\text{ m}\), distance from second pole = \(60\text{ m}\)

📅 CBSE 2026 (30/2/2)⭐ 2 Marks📝 SA-I

Q49. Find the angle of elevation of the sun when the shadow of a vertical pole of height \(h\) is \(\frac{h}{\sqrt{3}}\).

Solution: \(60^\circ\)

📅 CBSE 2024 (30/1/1)⭐ 2 Marks📝 SA-I

Q50. A contractor plans to install a slide for children in a park. For children below 5 years, she prefers a slide whose top is at a height of \(1.5\text{ m}\) and inclined at \(30^\circ\) to the ground. Find the length of this slide.

Solution: \(3\text{ m}\)

📅 CBSE 2024 (30/2/1)⭐ 3 Marks📝 SA-II

Q51. From the top of a \(7\text{ m}\) high building, the angle of elevation of the top of a cable tower is \(60^\circ\) and the angle of depression of its foot is \(45^\circ\). Determine the height of the tower.

Solution: \(7(\sqrt{3}+1)\text{ m}\)

📅 CBSE 2025 (30/1/3)⭐ 3 Marks📝 SA-II

Q52. The angle of elevation of the top of a building from the foot of a tower is \(30^\circ\) and the angle of elevation of the top of the tower from the foot of the building is \(60^\circ\). If the tower is \(60\text{ m}\) high, find the height of the building.

Solution: \(20\text{ m}\)

📅 CBSE 2024 (30/1/1)⭐ 3 Marks📝 SA-II

Q53. As observed from the top of a \(75\text{ m}\) high lighthouse from the sea-level, the angles of depression of two ships are \(30^\circ\) and \(45^\circ\). If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

Solution: \(75(\sqrt{3}-1)\text{ m}\)

📅 CBSE 2023 (30/3/1)⭐ 3 Marks📝 SA-II

Q54. Two poles of equal heights are standing opposite each other on either side of the road, which is \(80\text{ m}\) wide. From a point between them on the road, the angles of elevation of the top of the poles are \(60^\circ\) and \(30^\circ\), respectively. Find the height of the poles.

Solution: Height of poles = \(20\sqrt{3}\text{ m}\)

📅 CBSE 2026 (30/2/2)⭐ 3 Marks📝 SA-II

Q55. From the top of a \(10\text{ m}\) high building, the angle of elevation of the top of a tower is \(60^\circ\) and the angle of depression of its foot is \(45^\circ\). Find the height of the tower.

Solution: \(10(\sqrt{3}+1)\text{ m}\)

📅 CBSE 2023 (30/3/1)⭐ 3 Marks📝 SA-II

Q56. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of \(30^\circ\), which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be \(60^\circ\). Find the time taken by the car to reach the foot of the tower from this point.

Solution: \(3\) seconds

📅 CBSE 2020 (30/1/1)⭐ 3 Marks📝 SA-II

Q57. The angles of elevation of the top of a tower from two points at a distance of \(4\text{ m}\) and \(9\text{ m}\) from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is \(6\text{ m}\).

Solution: Proof completed. Let height be \(h\). \(\tan \theta = h/4\) and \(\tan(90-\theta) = h/9\). Multiplying both gives \(h^2/36 = 1 \implies h = 6\text{ m}\).

📅 CBSE 2025 (30/1/1)⭐ 3 Marks📝 SA-II

Q58. From a point on the ground, the angle of elevation of the top of a vertical tower is \(30^\circ\). On walking \(30\text{ m}\) towards the tower, the angle of elevation becomes \(60^\circ\). Find the height of the tower.

Solution: \(15\sqrt{3}\text{ m}\)

📅 CBSE 2024 (30/2/3)⭐ 3 Marks📝 SA-II

Q59. A \(1.5\text{ m}\) tall boy is standing at some distance from a \(30\text{ m}\) tall building. The angle of elevation from his eyes to the top of the building increases from \(30^\circ\) to \(60^\circ\) as he walks towards the building. Calculate the distance he walked.

Solution: \(19\sqrt{3}\text{ m}\)

📅 CBSE 2023 (30/3/2)⭐ 3 Marks📝 SA-II

Q60. A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is \(60^\circ\). From another point \(20\text{ m}\) away from this point on the line joining it to the foot of the tower, the angle of elevation of the top of the tower is \(30^\circ\). Find the height of the tower.

Solution: Height of tower = \(10\sqrt{3}\text{ m}\)

📅 CBSE 2026 (30/2/1)⭐ 3 Marks📝 SA-II

Q61. A statue, \(1.6\text{ m}\) tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is \(60^\circ\) and from the same point, the angle of elevation of the top of the pedestal is \(45^\circ\). Find the height of the pedestal.

Solution: \(0.8(\sqrt{3}+1)\text{ m}\)

📅 CBSE 2024 (30/1/2)⭐ 3 Marks📝 SA-II

Q62. A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height \(6\text{ m}\). At a point on the plane, the angle of elevation of the bottom and the top of the flagstaff are \(30^\circ\) and \(60^\circ\) respectively. Find the height of the tower.

Solution: Height of tower = \(3\text{ m}\)

📅 CBSE 2024 (30/2/3)⭐ 3 Marks📝 SA-II

Q63. A round balloon of radius \(r\) subtends an angle \(\alpha\) at the eye of the observer while the angle of elevation of its center is \(\beta\). Prove that the height of the center of the balloon is \(r \sin \beta \csc \frac{\alpha}{2}\).

Solution: Proof completed based on geometry of balloon radius, tangents, and right triangle trigonometry.

📅 CBSE 2025 (30/1/3)⭐ 3 Marks📝 SA-II

Q64. The angle of elevation of an aeroplane from a point on the ground is \(60^\circ\). After a flight of 30 seconds, the angle of elevation becomes \(30^\circ\). If the aeroplane is flying at a constant height of \(3000\sqrt{3}\text{ m}\), find the speed of the aeroplane.

Solution: Speed = \(200\text{ m/s}\) (or \(720\text{ km/h}\))

📅 CBSE 2025 (30/2/2)⭐ 3 Marks📝 SA-II

Q65. Two boats are approaching a lighthouse from opposite directions. The angle of depression of the boats from the top of the \(100\text{ m}\) high lighthouse are \(30^\circ\) and \(45^\circ\). Find the distance between the two boats.

Solution: \(100(\sqrt{3}+1)\text{ m}\)

📅 CBSE 2024 (30/1/1)⭐ 5 Marks📝 LA

Q66. The angle of elevation of a cloud from a point \(60\text{ m}\) above a lake is \(30^\circ\) and the angle of depression of the reflection of the cloud in the lake is \(60^\circ\). Find the height of the cloud from the surface of the lake.

Solution: Height of cloud = \(120\text{ m}\)

📅 CBSE 2024 (30/1/2)⭐ 5 Marks📝 LA

Q67. From the top of a \(75\text{ m}\) high lighthouse, the angles of depression of two ships approaching it are \(30^\circ\) and \(60^\circ\). If one ship is directly behind the other on the same side of the lighthouse, find the distance between the two ships.

Solution: \(50\sqrt{3}\text{ m}\)

📅 CBSE 2025 (30/1/3)⭐ 5 Marks📝 LA

Q68. Two poles of equal heights are standing opposite each other on either side of the road, which is \(100\text{ m}\) wide. From a point on the road between the poles, the angles of elevation of the top of the poles are \(60^\circ\) and \(30^\circ\). Find the height of the poles and the distance of the point from the poles.

Solution: Height of poles = \(25\sqrt{3}\text{ m}\), distances are \(25\text{ m}\) and \(75\text{ m}\).

📅 CBSE 2017 (30/1)⭐ 5 Marks📝 LA

Q69. The angle of elevation of the top of a tower from certain point is \(30^\circ\). If the observer moves \(20\text{ m}\) towards the tower, the angle of elevation of the top increases by \(15^\circ\). Find the height of the tower.

Solution: \(10(\sqrt{3}+1)\text{ m}\)

📅 CBSE 2024 (30/2/1)⭐ 5 Marks📝 LA

Q70. A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height \(5\text{ m}\). At a point on the plane, the angles of elevation of the bottom and top of the flagstaff are \(30^\circ\) and \(60^\circ\) respectively. Find the height of the tower.

Solution: Height of tower = \(2.5\text{ m}\)

📅 CBSE 2024 (30/2/1)⭐ 4 Marks📝 Case Study

Q71. Case Study 1: Drone Photography and Mapping
A municipal corporation uses a high-definition mapping drone to create a 3D model of a newly planned Smart City square. The drone is programmed to hover at a constant height of \(120\text{ m}\) vertically above a central junction point \(O\) on the ground.

Based on the trigonometric mapping parameters:

1. If the drone cameras spot a landmark \(A\) at an angle of depression of \(45^\circ\), find the horizontal distance \(OA\). (1 Mark)
2. At the same instant, another landmark \(B\) is spotted on the opposite side of \(OA\) such that the angle of depression of \(B\) is \(30^\circ\). Find the horizontal distance \(OB\). (1 Mark)
3. Find the direct line-of-sight distance from the drone to landmark \(B\). Or, find the straight line distance \(AB\) between the two landmarks if they are collinear with \(O\). (2 Marks)

Solution:
1. \(120\text{ m}\)
2. \(120\sqrt{3}\text{ m}\)
3. \(240\text{ m}\) OR \(120(\sqrt{3}+1)\text{ m}\)

📅 CBSE 2022 (30-2-3)⭐ 4 Marks📝 Case Study

Q72. Case Study 2: Hot Air Balloon Flight Paths
During a tourist festival in Rajasthan, a hot air balloon is released from a point on the ground. An observer at point \(P\), standing at a distance of \(50\text{ m}\) from the launch pad, watches the balloon rise vertically.

Based on the flight telemetry:

1. Write the trigonometric equation when the angle of elevation of the balloon is \(30^\circ\). (1 Mark)
2. Find the height of the balloon when the angle of elevation is \(45^\circ\). (1 Mark)
3. If the balloon rises further such that the angle of elevation becomes \(60^\circ\), find the vertical distance traveled by the balloon between \(45^\circ\) and \(60^\circ\). Or, find the line-of-sight distance from observer \(P\) to the balloon when the angle of elevation is \(60^\circ\). (2 Marks)

Solution:
1. \(h/50 = \tan 30^\circ\)
2. \(50\text{ m}\)
3. \(50(\sqrt{3}-1)\text{ m}\) OR \(100\text{ m}\)

📅 CBSE 2024 (30/2/2)⭐ 4 Marks📝 Case Study

Q73. Case Study 3: The Lighthouse Safety Beacon
A lighthouse beacon sits on a high cliff overlooking a busy shipping lane. The lighthouse keeper observes two merchant vessels approaching the port. The height of the beacon from the sea level is \(150\text{ m}\).

Based on the lighthouse observation angles:

1. If the keeper notes the angle of depression of ship \(A\) as \(30^\circ\), what is the horizontal distance of ship \(A\) from the base of the cliff? (1 Mark)
2. If ship \(B\) is closer to the lighthouse and its angle of depression is \(45^\circ\), find the horizontal distance of ship \(B\) from the base of the cliff. (1 Mark)
3. Calculate the distance between ship \(A\) and ship \(B\) if they are in the same straight line with the lighthouse base. Or, find the distance between the ships if ship \(B\) is on the opposite side of the lighthouse base. (2 Marks)

Solution:
1. \(150\sqrt{3}\text{ m}\)
2. \(150\text{ m}\)
3. \(150(\sqrt{3}-1)\text{ m}\) OR \(150(\sqrt{3}+1)\text{ m}\)

📅 CBSE 2025 (30/1/3)⭐ 4 Marks📝 Case Study

Q74. Case Study 4: Multi-Storey Building and Tower
From the top of a \(50\text{ m}\) high multi-storey building, the angles of depression of the top and bottom of a vertical transmission tower are observed to be \(30^\circ\) and \(60^\circ\), respectively.

Based on the architectural measurements:

1. Find the horizontal distance between the building and the transmission tower. (1 Mark)
2. Represent the scenario using a simplified trigonometric right-triangle equation for the building. (1 Mark)
3. Calculate the height of the transmission tower. Or, find the line-of-sight distance from the top of the building to the bottom of the tower. (2 Marks)

Solution:
1. \(\frac{50}{\sqrt{3}}\text{ m}\)
2. \(50/x = \tan 60^\circ\)
3. \(33.33\text{ m}\) OR \(\frac{100}{\sqrt{3}}\text{ m}\)

📅 CBSE 2024 (30/1/1)⭐ 4 Marks📝 Case Study

Q75. Case Study 5: Meteorological Balloon Observation
A meteorological station releases a helium weather balloon to measure atmospheric pressure and wind speed. Two observation stations, \(A\) and \(B\), on a straight road \(1000\text{ m}\) apart, track the balloon at the same instant.

Based on the tracking data:

1. If the angles of elevation of the balloon from stations \(A\) and \(B\) are \(45^\circ\) and \(60^\circ\) respectively, and the balloon is hovering in the vertical plane between \(A\) and \(B\), write down the equation relating the heights and horizontal distances. (1 Mark)
2. Find the height of the balloon above the ground. (1 Mark)
3. If the balloon moves horizontally towards station \(A\) by \(100\text{ m}\), find its new angle of elevation from station \(A\) (keeping height constant). Or, calculate the distance of the balloon from station \(B\) along the line-of-sight. (2 Marks)

Solution:
1. \(h\cot 45^\circ + h\cot 60^\circ = 1000\)
2. \(500(3-\sqrt{3})\text{ m}\)
3. \(\tan \theta \approx 0.65\) OR \(\frac{2h}{\sqrt{3}}\)

Live Practice: Chapter 9 Applications of Trigonometry

Q 1

MCQ Score: 0/0